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Lecturer(s)
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Mikeš Josef, prof. RNDr. DrSc.
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Peška Patrik, RNDr. Ph.D.
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Course content
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1. n-dimensional differentiable manifolds. 2. Geometric objects on manifolds. 3. Tensors on manifolds. 4. Manifolds with affine connection, covariant derivation. 5. Parallel transport. Geodetic curves. 6. Riemannian and Ricci tensors. 7. Riemannian metrics, length of curves. 8. Variation problems on manifolds. 9. Geodetic curves on Riemannian space. 10. Properties of Riemannian and Ricci tensors. 11. Sectional curvature on Riemannian space. 12. Spaces on constant curvature, Einstein spaces. 13. Isometric and conformal mappings.
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Learning activities and teaching methods
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Lecture, Demonstration, Grafic and Art Activities
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Learning outcomes
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Understand basic topics of differential and integral calculus on manifolds.
Comprehension of the theory of curves, surfaces and theirn higher-dimensional generalization, ability to use them in physical and technical applications.
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Prerequisites
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unspecified
KAG/GEO1M
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Assessment methods and criteria
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Student performance
Active participation.
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Recommended literature
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Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. VUT Brno.
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Eisenhart, L.P. (2000). Non-Riemannian Geometry. Amer. Math. Soc. Colloquium Publ. 8.
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J. Mikeš, E. Stepanova, A. Vanžurová et al. (2015). Differential geometry of special mappings. UP Olomouc.
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J. Mikeš, M. Sochor. (2013). Diferenciální geometrie ploch v úlohách. UP OLomouc.
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J. Mikeš, P. Peška et al. (2015). Differential geometry of special mappings. UP Olomouc.
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Kowalski, O. (1995). Úvod do Riemannovy geometrie. Praha.
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Oprea, J. (2007). Differential geometry and its aplications. MAA Pearson Educ.
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Pogorelov, A. V. (1969). Diferencialnaja geometrija.. Nauka Moskva.
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